Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow related to quantum physics. But when I read the statement of this theorem on Wikipedia, I immediately came to think of a possible application to general relativity. So, considering that space-time is locally isomorphic to $\mathbb{C}^{2}$, can this theorem explain why there should not be any naked singularity?
Sorry if the question is rather vague.
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asked May 24, 2013 at 21:52
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1 Answer
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Hartog's theorem states that on $\mathbb{C}^n$, $n\ge 2$, if $f$ is a holomorphic function on $G\backslash K$ where $G\subset \mathbb{C}^n$ is open, $K$ is compact and $G\backslash K$ is connected then $f$ can be holomorphically extended to the whole $G$. In other words, singularities cannot stay bounded in compact sets.
So the anwer is No, for two reasons. The first is that the metric components in general relativity are $C^2$, hence in general not holomorphic. The second is that, in any case, naked singularities are expected, mathematically speaking, at infinity: a curve headed at the singularity escapes every compact set. Think for instance of 1+1 Minkowski spacetime with the usual coordinates (t,x) and with the set $x\le 0$ removed. The line $x=0$ is made by naked singularity points.
